Probability of decoding error for random phase and Rayleigh fading channels

In this paper, the probability of error of two time-varying channels with memory, 1) the random phase channel and 2) the Rayleigh fading channel, is discussed. The input is assumed to be one of M equiprobable waveforms. An upper bound to the probability of error is derived by convetting the channel into a memoryless one by means of scrambling. A lower bound to the probability of error is derived by assuming that except for an additive noise, all the channel parameters are completely known at the receiver and, therefore, are not considered to be random variables any more. Following this assumption, it is shown that the channel is converted into a memoryless channel. For each one of the two channels, there is a region of SNR where the upper and lower bounds are close together and therefore yield a good estimate to the actual probability of error. Furthermore, using random coding and scrambling, this probability of error may actually be achieved (for a certain region of rates).